Theorem 19.26 1481

Description: Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)

Assertion

Ref	Expression
19.26	⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))

Proof of Theorem 19.26

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2	1	alimi 1455	. . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → ∀𝑥𝜑)
3		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
4	3	alimi 1455	. . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → ∀𝑥𝜓)
5	2, 4	jca 306	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) → (∀𝑥𝜑 ∧ ∀𝑥𝜓))
6		id 19	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓))
7	6	alanimi 1459	. 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥(𝜑 ∧ 𝜓))
8	5, 7	impbii 126	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wal 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.26-2  1482  19.26-3an  1483  albiim  1487  2albiim  1488  hband  1489  hban  1547  19.27h  1560  19.27  1561  19.28h  1562  19.28  1563  nford  1567  nfand  1568  equsexd  1729  equveli  1759  sbanv  1889  2eu4  2119  bm1.1  2162  r19.26m  2608  unss  3310  ralunb  3317  ssin  3358  intun  3876  intpr  3877  eqrelrel  4728  relop  4778  eqoprab2b  5933  dfer2  6536  omniwomnimkv  7165